Circle Arc and Sector Calculator
Calculate the arc length, sector area, or circular segment area for a given radius and central angle in degrees or radians. Perfect for geometry problems, engineering layouts, and design work.
About this calculator
A central angle θ (in radians) cuts out a portion of a circle of radius r. Arc length is the curved distance along the circle's edge: L = rθ. Sector area is the pie-slice region bounded by two radii and the arc: A_sector = (1/2)r²θ. Segment area is the region between a chord and its arc: A_segment = (1/2)r²(θ − sin θ). If your angle is in degrees, convert first: θ_rad = θ_deg × π/180. These formulas are essential in engineering (cam profiles, road curves), navigation, and any application requiring precise circular geometry.
How to use
Suppose r = 10 cm and the central angle is 60°. Convert: θ = 60 × π/180 = π/3 ≈ 1.0472 rad. Arc length: L = 10 × 1.0472 ≈ 10.47 cm. Sector area: A = 0.5 × 10² × 1.0472 = 0.5 × 100 × 1.0472 ≈ 52.36 cm². Segment area: A = 0.5 × 100 × (1.0472 − sin(60°)) = 50 × (1.0472 − 0.8660) = 50 × 0.1812 ≈ 9.06 cm². Enter radius = 10, angle = 60, unit = degrees, select your calculation type, and the result appears immediately.
Frequently asked questions
What is the difference between a sector and a segment of a circle?
A sector is the 'pie slice' region enclosed by two radii and the arc between them, like a slice of pizza. A segment is the region between a chord (the straight line connecting two points on the circle) and the arc it subtends. The segment area equals the sector area minus the triangular area formed by the two radii and the chord, which is why A_segment = (1/2)r²(θ − sin θ). Segments appear frequently in problems involving cross-sections of cylinders or water levels in tanks.
How do you convert degrees to radians for arc and sector calculations?
Radians and degrees are two ways to measure angles; a full circle is 2π radians or 360 degrees. To convert degrees to radians, multiply by π/180: θ_rad = θ_deg × π/180. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 rad. The radian measure is used directly in the arc and sector formulas because it represents the ratio of arc length to radius, making the formulas L = rθ and A = (1/2)r²θ dimensionally natural.
When would you need to calculate arc length in real life?
Arc length calculations appear whenever you need the actual curved distance along a circular path rather than the straight-line chord distance. Road engineers calculate arc lengths when designing curved highway sections or roundabouts. Machinists use arc length when programming CNC machines to cut circular profiles. In athletics, the length of a curved running lane on a track is an arc length calculation. Any situation involving cables, belts, or materials that wrap around a circular object also requires arc length.